Fact: Start with $V$ a subspace of $\mathbb R^n$. Take the set of all supports of vectors in $V$. Throw out $\emptyset$. You now have the dependent sets of some matroid.
Not sure you believe me? Or just want to get your hands on the independent sets? (If you tell people that you have a matroid, they always ask you for the independent sets. How can you blame them?) Okay, no problem.
Choose a basis for $V^\perp$ and take those vectors as the rows of a matrix $A$. The sets of indices corresponding to linearly independent columns give the independent sets of the matroid. (A set of indices corresponds to linearly dependent columns, of course, exactly when it can be matched up with nonzero scalars so as to give a vector orthogonal to every row, i.e., a vector in $V$. This also shows that it doesn't matter which basis you choose.)
It follows from the above that the matroids that arise in this fashion are precisely the ones that are representable over $\mathbb R$.
I like this way of thinking about representability. But I've never seen it presented this way in the literature. Why not? (Maybe it has to do with how I seem to be relying on a nice Euclidean inner product?) This leads me to a couple of questions.
- Is the fact I stated above presented in any reference anywhere? I would greatly appreciate it if anyone could point me in the direction of one.
- Is it the case that the fact I stated above remains true when $\mathbb R$ is replaced by any field? If so, does it characterize representability over any field?
EDIT: I just realized that it's not difficult to prove that the above fact does hold over any field; I believe one can simply show that those support sets which are minimal with respect to $\subseteq$ satisfy the circuit axioms, and that's enough. So my main question here is whether or not this stills serves to characterize representability over other fields.